Akhmedov, Anar; Ozbagci, Burak Exotic Stein fillings with arbitrary fundamental group. (English) Zbl 1397.57045 Geom. Dedicata 195, 265-281 (2018). A \(4\)-manifold \(X\) is said to be a Stein filling of a closed contact \(3\)-manifold \((M, \xi)\) if \(X\) is the sub-level set of a plurisubharmonic function on a Stein surface and the contact structure induced on \(\partial X\) by complex tangencies is contactomorphic to \((M, \xi)\). Two Stein fillings are said to be exotic if they are homeomorphic but non-diffeomorphic. A group \(G\) is said to be finitely presentable if it possesses a presentation \(\langle x_1,\dots,x_k\mid r_1,\dots,r_s\rangle\) with finitely many generators, and finitely many relations. In [J. Differ. Geom. 53, No. 2, 205–236 (1999; Zbl 1040.53094)], S. K. Donaldson proved that every closed symplectic 4-manifold admits a Lefschetz pencil over \(S^2\) and R. E. Gompf in [Ann. Math. (2) 142, No. 3, 527–595 (1995; Zbl 0849.53027)] showed that every finitely presentable group \(G\) can be realized as the fundamental group of some closed symplectic 4-manifold.In this paper, the authors prove that for any finitely presentable group \(G\) there exists an infinite family of homeomorphic but pairwise non-diffeomorphic (exotic) symplectic but non-complex closed 4-manifolds with fundamental group \(G\) such that each member of this family admits a Lefschetz fibration of the same genus over \(S^2\). Also, they show that for any finitely presentable group \(G\) there exists a contact 3-manifold \(M\) which admits infinitely many homeomorphic but pairwise non-diffeomorphic Stein fillings such that the fundamental group of each filling is isomorphic to \(G\). Moreover, it is shown that the contact 3-manifold \(M\) is contactomorphic to the link of some isolated complex surface singularity equipped with its canonical contact structure \(\xi_{\text{sd}}\). Reviewer: Andrew Bucki (Edmond) Cited in 6 Documents MSC: 57R17 Symplectic and contact topology in high or arbitrary dimension Keywords:Lefschetz fibrations; Stein fillings; contact structures; exotic manifolds; symplectic manifolds Citations:Zbl 1040.53094; Zbl 0849.53027 PDFBibTeX XMLCite \textit{A. Akhmedov} and \textit{B. Ozbagci}, Geom. Dedicata 195, 265--281 (2018; Zbl 1397.57045) Full Text: DOI arXiv References: [1] Akaho, M, A connected sums of knots and fintushel-stern knot surgery, Turk. J. Math., 30, 87-93, (2006) · Zbl 1095.57026 [2] Akbulut, S; Ozbagci, B, Lefschetz fibrations on compact Stein surfaces, Geom. Topol., 5, 319-334, (2001) · Zbl 1002.57062 [3] Akhmedov, A, Construction of symplectic cohomology \({\mathbb{S}}^{2}× {\mathbb{S}}^{2}\), Gökova Geom. Topol. Proc., 14, 36-48, (2007) · Zbl 1189.53080 [4] Akhmedov, A; Baykur, Rİ; Park, BD, Constructing infinitely many smooth structures on small 4-manifolds, J. 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